Isoperimetric inequalities under bounded integral norms of Ricci curvature and mean curvature

Paeng, Seong-Hun

doi:10.1090/proc/13813

Isoperimetric inequalities under bounded integral norms of Ricci curvature and mean curvature
HTML articles powered by AMS MathViewer

by Seong-Hun Paeng PDF

Proc. Amer. Math. Soc. 146 (2018), 1309-1323 Request permission

Abstract:

We obtain isoperimetric inequalities under bounded integral norms of Ricci curvature and mean curvature. Also we generalize the results to metric-measure spaces.

References

D. Bakry and Michel Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206 (French). MR 889476, DOI 10.1007/BFb0075847
Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635
Isaac Chavel, Riemannian geometry—a modern introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. MR 1271141
Isaac Chavel, Isoperimetric inequalities, Cambridge Tracts in Mathematics, vol. 145, Cambridge University Press, Cambridge, 2001. Differential geometric and analytic perspectives. MR 1849187
X. Dai, G. Wei, and Z. Zhang, Local Sobolev constant estimate for integral Ricci curvature bounds, arXiv:1601.0819 (2016).
Sylvestre Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Astérisque 157-158 (1988), 191–216. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). MR 976219
Gregory J. Galloway and Eric Woolgar, Cosmological singularities in Bakry-Émery spacetimes, J. Geom. Phys. 86 (2014), 359–369. MR 3282334, DOI 10.1016/j.geomphys.2014.08.016
Kit Hanes, Isoperimetric inequalities for manifolds with boundary, J. Differential Geometry 7 (1972), 525–534. MR 334081
David Hoffman and Joel Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715–727. MR 365424, DOI 10.1002/cpa.3160270601
Chuan-chih Hsiung, Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. of Math. (2) 73 (1961), 213–220. MR 130637, DOI 10.2307/1970287
John Lott, Some geometric properties of the Bakry-Émery-Ricci tensor, Comment. Math. Helv. 78 (2003), no. 4, 865–883. MR 2016700, DOI 10.1007/s00014-003-0775-8
Seong-Hun Paeng, Buser’s isoperimetric inequalities with integral norms of Ricci curvature, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2903–2910. MR 2801631, DOI 10.1090/S0002-9939-2010-10725-6
Seong-Hun Paeng, Volumes and intrinsic diameters of hypersurfaces, J. Geom. Phys. 95 (2015), 96–107. MR 3357824, DOI 10.1016/j.geomphys.2015.05.001
P. Petersen and G. Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997), no. 6, 1031–1045. MR 1487753, DOI 10.1007/s000390050036
William T. Reid, The isoperimetric inequality and associated boundary problems, J. Math. Mech. 8 (1959), 897–905. MR 0130623, DOI 10.1512/iumj.1959.8.58057
Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377–405. MR 2577473, DOI 10.4310/jdg/1261495336